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In [[mathematics]], the  '''Jucys&ndash;Murphy elements''' in the [[group algebra]] <math>\mathbb{C} [S_n] </math> of the [[symmetric group]], named after [[Algimantas Adolfas Jucys]] and G. E. Murphy, are defined as a sum of [[transposition (mathematics)|transposition]]s by the formula:


:<math>X_1=0, ~~~  X_k= (1 k)+ (2 k)+\cdots+(k-1\ k), ~~~ k=2,\dots,n. </math>


They play an important role in the [[representation theory]] of the [[symmetric group]].
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==Properties==
They generate a commutative subalgebra of <math>\mathbb{C} [ S_n] </math>. Moreover,
''X''<sub>''n''</sub> commutes with all elements of <math>\mathbb{C} [S_{n-1}] </math>.
 
The vectors of the Young basis are eigenvectors for the action of ''X''<sub>''n''</sub>. For any [[standard Young tableau]] ''U'' we have:
 
:<math>X_k v_U =c_k(U) v_U, ~~~ k=1,\dots,n, </math>
 
where ''c''<sub>''k''</sub>(''U'') is the ''content'' ''b''&nbsp;&minus;&nbsp;''a'' of the cell (''a'',&nbsp;''b'') occupied by  ''k'' in the standard Young tableau&nbsp;''U''.
 
'''Theorem''' (Jucys): The [[Center (algebra)|center]] <math>Z(\mathbb{C} [S_n])</math> of the group algebra <math>\mathbb{C} [S_n] </math> of the symmetric group is generated by the [[symmetric polynomial]]s in the elements ''X<sub>k</sub>''.
 
'''Theorem''' (Jucys): Let ''t'' be a formal variable commuting with everything, then the following identity for polynomials in variable ''t'' with values in the group algebra  <math>\mathbb{C} [S_n] </math> holds true:
 
:<math> (t+X_1) (t+X_2) \cdots (t+X_n)= \sum_{\sigma \in S_n} \sigma t^{\text{number of cycles of }\sigma}.</math>
 
'''Theorem''' ([[Okounkov]]&ndash;[[Vershik]]): The subalgebra of  <math>\mathbb{C} [S_n] </math> generated by the centers
 
: <math> Z(\mathbb{C} [ S_1]), Z(\mathbb{C} [ S_2]), \ldots,  Z(\mathbb{C} [ S_{n-1}]),  Z(\mathbb{C} [S_n])  </math>
 
is exactly the subalgebra generated by the Jucys&ndash;Murphy elements ''X<sub>k</sub>''.
 
==See also==
* [[Representation theory of the symmetric group]]
* [[Young symmetrizer]]
 
==References==
*{{Citation
|title=A New Approach to the Representation Theory of the Symmetric Groups. 2
|authorlink1=Okounkov
|first1=Andrei |last1=Okounkov
|authorlink2=Vershik
|first2=Anatoly |last2=Vershik
|year=2004
|volume=307
|journal=Zapiski Seminarod POMI (In Russian) v.
|arxiv = math.RT/0503040
|postscript=  (revised English version). }}
 
*{{citation
|title=Symmetric polynomials and the center of the symmetric group ring
|authorlink1=Algimantas Adolfas Jucys
|first1=Algimantas Adolfas  |last1=Jucys
| year=1974 | journal=Rep. Mathematical Phys. | volume=5 | issue=1  | pages=107–112
| doi=10.1016/0034-4877(74)90019-6 
}}
 
*{{citation
|title=On the Young operators of the symmetric group
|authorlink1=Algimantas Adolfas Jucys
|first1=Algimantas Adolfas  |last1=Jucys
| year=1966 | journal=Lietuvos Fizikos Rinkinys | volume=6  | pages=163–180
}}
 
*{{citation
|title=Factorization of Young projection operators for the symmetric group
|authorlink1=Algimantas Adolfas Jucys
|first1=Algimantas Adolfas  |last1=Jucys
| year=1971 | journal=Lietuvos Fizikos Rinkinys | volume=11  | pages=5–10
}}
 
*{{citation
|title=A new construction of Young's seminormal representation of the symmetric group
|first1=G. E. |last1=Murphy
| year=1981 | journal=J. Algebra | volume=69  | pages=287–297
|doi=10.1016/0021-8693(81)90205-2
|issue=2
}}
 
{{DEFAULTSORT:Jucys-Murphy Element}}
[[Category:Permutation groups]]
[[Category:Representation theory]]
[[Category:Symmetry]]
[[Category:Representation theory of finite groups]]
[[Category:Symmetric functions]]

Latest revision as of 12:51, 6 July 2014


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